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Don Bernard Zagier (born 29 June 1951) is an American-German mathematician whose main area of work is number theory . He is currently one of the directors of the Max Planck Institute for Mathematics in Bonn , Germany. He was a professor at the Collège de France in Paris from 2006 to 2014. Since October 2014, he is also a Distinguished Staff Associate at the International Centre for Theoretical Physics (ICTP ).[2]
Background
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Zagier was born in Heidelberg , West Germany . His mother was a psychiatrist, and his father was the dean of instruction at the American College of Switzerland . His father held five different citizenships, and he spent his youth living in many different countries. After finishing high school (at age 13) and attending Winchester College for a year, he studied for three years at MIT , completing his bachelor's and master's degrees and being named a Putnam Fellow in 1967 at the age of 16.[3] He then wrote a doctoral dissertation on characteristic classes under Friedrich Hirzebruch at Bonn , receiving his PhD at 20. He received his Habilitation at the age of 23, and was named professor at the age of 24.[4]
Work
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Zagier collaborated with Hirzebruch in work on Hilbert modular surfaces . Hirzebruch and Zagier coauthored Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, [5] where they proved that intersection numbers of algebraic cycles on a Hilbert modular surface occur as Fourier coefficients of a modular form . Stephen Kudla , John Millson and others generalized this result to intersection numbers of algebraic cycles on arithmetic quotients of symmetric spaces.[6]
One of his results is a joint work with Benedict Gross (the so-called Gross–Zagier formula ). This formula relates the first derivative of the complex L-series of an elliptic curve evaluated at 1 to the height of a certain Heegner point . This theorem has some applications, including implying cases of the Birch and Swinnerton-Dyer conjecture , along with being an ingredient to Dorian Goldfeld 's solution of the class number problem . As a part of their work, Gross and Zagier found a formula for norms of differences of singular moduli.[7] Zagier later found a formula for traces of singular moduli as Fourier coefficients of a weight 3/2 modular form .[8]
Zagier collaborated with John Harer to calculate the orbifold Euler characteristics of moduli spaces of algebraic curves , relating them to special values of the Riemann zeta function .[7]
Zagier found a formula for the value of the Dedekind zeta function of an arbitrary number field at s = 2 in terms of the dilogarithm function, by studying arithmetic hyperbolic 3-manifolds .[9] He later formulated a general conjecture giving formulas for special values of Dedekind zeta functions in terms of polylogarithm functions.[10]
He discovered a short and elementary proof of Fermat's theorem on sums of two squares .[11] [12]
Zagier won the Cole Prize in Number Theory in 1987,[13] the Chauvenet Prize in 2000,[1] the von Staudt Prize in 2001[14] and the Gauss Lectureship of the German Mathematical Society in 2007. He became a foreign member of the Royal Netherlands Academy of Arts and Sciences in 1997[15] and a member of the National Academy of Sciences (NAS) of the United States in 2017.
Selected publications
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Zagier, D. (1990), "A One-Sentence Proof That Every Prime p ≡ 1 (mod 4 ) Is a Sum of Two Squares", The American Mathematical Monthly , 97 (2 ), Mathematical Association of America: 144, doi :10.2307/2323918 , JSTOR 2323918 . The First 50 Million Prime Numbers." Math. Intel. 0, 221–224, 1977.
Hirzebruch, F.; Zagier, D. (1976). "Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus". Inventiones Mathematicae . 36 (1 ). Springer Science and Business Media LLC: 57–113. Bibcode :1976InMat..36...57H . doi :10.1007/bf01390005 . hdl :21.11116/0000-0004-399B-E . ISSN 0020-9910 . S2CID 56568473 .
Zagier, Don (1986). "Hyperbolic manifolds and special values of Dedekind zeta-functions". Inventiones Mathematicae . 83 (2 ). Springer Science and Business Media LLC: 285–301. Bibcode :1986InMat..83..285Z . doi :10.1007/bf01388964 . ISSN 0020-9910 . S2CID 67757648 .
"On singular moduli". Journal für die reine und angewandte Mathematik (Crelle's Journal) . 1985 (355). Walter de Gruyter GmbH: 191–220. 1 January 1985. doi :10.1515/crll.1985.355.191 . ISSN 0075-4102 . S2CID 117887979 .
Gross, Benedict H.; Zagier, Don B. (1986). "Heegner points and derivatives of L-series". Inventiones Mathematicae . 84 (2 ). Springer Science and Business Media LLC: 225–320. Bibcode :1986InMat..84..225G . doi :10.1007/bf01388809 . ISSN 0020-9910 . S2CID 125716869 .
Harer, J.; Zagier, D. (1986). "The Euler characteristic of the moduli space of curves". Inventiones Mathematicae . 85 (3 ). Springer Science and Business Media LLC: 457–485. arXiv :math/0506083 . Bibcode :1986InMat..85..457H . doi :10.1007/bf01390325 . ISSN 0020-9910 . S2CID 8471412 .
Gross, B.; Kohnen, W.; Zagier, D. (1987). "Heegner points and derivatives of L-series. II". Mathematische Annalen . 278 (1–4). Springer Science and Business Media LLC: 497–562. doi :10.1007/bf01458081 . ISSN 0025-5831 . S2CID 121652706 .
Zagier, Don (1991). "The Birch-Swinnerton-Dyer Conjecture from a Naive Point of View". Arithmetic Algebraic Geometry . Boston, MA: Birkhäuser Boston. pp. 377–389. doi :10.1007/978-1-4612-0457-2_18 . ISBN 978-1-4612-6769-0 .
Zagier, Don (1991). "Polylogarithms, Dedekind Zeta Functions, and the Algebraic K-Theory of Fields". Arithmetic Algebraic Geometry . Boston, MA: Birkhäuser Boston. pp. 391–430. doi :10.1007/978-1-4612-0457-2_19 . ISBN 978-1-4612-6769-0 .
Zagier, Don (1990). "How Often Should You Beat Your Kids?". Mathematics Magazine . 63 (2 ). Informa UK Limited: 89–92. doi :10.1080/0025570x.1990.11977493 . ISSN 0025-570X .
See also
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References
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^ "Putnam Competition Individual and Team Winners" . Mathematical Association of America . Retrieved December 13, 2021 .
^ "Don Zagier" . Max Planck Institute for Mathematics . Retrieved 19 November 2020 .
^ Hirzebruch, Friedrich ; Zagier, Don (1976). "Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus". Inventiones Mathematicae . 36 : 57–113. Bibcode :1976InMat..36...57H . doi :10.1007/BF01390005 . hdl :21.11116/0000-0004-399B-E . S2CID 56568473 .
^ Kudla, Stephen S. (1997). "Algebraic cycles on Shimura varieties of orthogonal type" . Duke Mathematical Journal . 86 (1 ): 39–78. doi :10.1215/S0012-7094-97-08602-6 . Archived from the original on March 3, 2016 – via Project Euclid and Wayback Machine .
^ a b Harer, J.; Zagier, D. (1986). "The Euler characteristic of the moduli space of curves" (PDF) . Inventiones Mathematicae . 85 (3 ): 457–485. Bibcode :1986InMat..85..457H . doi :10.1007/BF01390325 . S2CID 17634229 .
^ Zagier, Don (1985). "TRACES OF SINGULAR MODULI". J. Reine Angew. Math . CiteSeerX 10.1.1.453.3566 .
^ Zagier, Don (1986). "Hyperbolic manifolds and special values of Dedekind zeta-functions" (PDF) . Inventiones Mathematicae . 83 (2 ): 285–301. Bibcode :1986InMat..83..285Z . doi :10.1007/BF01388964 . S2CID 67757648 .
^ Zagier, Don . "Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields" (PDF) .
^ Snapper, Ernst (1990). "Inverse Functions and their Derivatives" . The American Mathematical Monthly . 97 (2 ): 144–147. doi :10.1080/00029890.1990.11995566 .
^ "One-Sentence Proof That Every Prime p congruent to 1 modulo 4 Is a Sum of Two Squares" . math.unh.edu . Archived from the original on 2012-02-05.
^ Frank Nelson Cole Prize in Number Theory , American Mathematical Society . Accessed March 17, 2010
^ Zagier Receives Von Staudt Prize. Notices of the American Mathematical Society , vol. 48 (2001), no. 8, pp. 830–831
^ "D.B. Zagier" . Royal Netherlands Academy of Arts and Sciences. Archived from the original on 14 February 2016. Retrieved 14 February 2016 .
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